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Question 6 2011 (a) Prove that every continuous function f : [0, 1] → [0, 1] has a fixed point, that is, the equation f(x) = x has at least one solution x ∈ [0, 1]. (b) Give an example of a continuous function f :]0, 1] →]0, 1] which does not have a fixed point. (c) Prove that no continuous surjective function f :]0, 1] → R can be injective. Question 4 2009 (a) Show that the topological space (X, T) is Hausdorff (T2) if and only if the diagonal (x, x)|x ∈ X is closed in X × X with the product topology. (b) Let (X, T) and (Y, U) be topological spaces. Prove that f : X → Y is continuous if and only if for each A ⊆ X, f(A¯) ⊆ f(A). Question 7 2010 (a) Prove that a compact Hausdorff space is normal. (b) Prove that there is no continuous bijection f : [0, 1] → [0, 1[. Question 6 2010 Decide which of the following statements are true and which are false. Provide a proof for those which are true. Give a counter-example for those which are false. - A closed subset of a compact space is compact. - Every contracting map f :]0, 1] →]0, 1] has a unique fixed point, where 0, 1] is taken with its Euclidean metric. Question 1 2010 Let X be the set of all absolutely convergent series of real numbers, so that X = {(xn)n∈N|xn ∈ R for all n ∈ N and X n∈N |xn| converges}. Decide whether ρ : X × X → R + 0 ((xn)n∈N,(yn)n∈N) 7−→ sup n∈N |xn − yn| determines a metric on X. Carefully justify your answer.

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